Method and apparatus for detecting angular position and amount of dynamic unbalance of rotating body

ABSTRACT

A method and apparatus for detecting the angular position and amount of a dynamic unbalance of a rotary body by detecting the direction and magnitude of a displacement of the normal axis of rotation of a rotationally vibrated rotary body, the displacement corresponding to the position and amount of the unbalance of said rotary body. The rotary body is rotationally vibrated within a minute angular range around its own normal axis of rotation and is resiliently supported to two spaced points along said axis so that it may be displaced in a plane perpendicular to said axis. As a result, said normal axis of rotation of the rotary body is displaced as a function of the angular position and amount of the unbalance of the rotary body. The magnitude and direction of the displacement are detected, whereby the angular position and amount of the dynamic unbalance of the rotary body are detected.

[mite States Patent [1 1 Sakuraba METHOD AND APPARATUS FOR DETECTINGANGULAR POSITION AND AMOUNT OF DYNAMIC UNBALANCE OF ROTATING BODY [75]Inventor: I-Iirosi Sakuraba, Kanagawa-ken,

Japan [73] Assignee: International Mechanical Vibration Laboratory,Inc., Osaka-shi, Osaka, Japan [22] Filed: Aug. 13, 1971 [2]] Appl. No.:171,460

[52] U.S. Cl ..73/460, 73/461 [51] Int. Cl. ..G0lm l/l6 [58] Field ofSearch ..73/66, 459, 460, 461, 462

[56] References Cited UNITED STATES PATENTS 3,130,576 '4/l964 Giers eta1. ..73/66 1March 13, 1973 Primary Examiner-James J. GillAtt0mey-R0bert D. Flynn et al.

[57] ABSTRACT A method and apparatus for detecting the angular positionand amount of a dynamic unbalance of a rotary body by detecting thedirection and magnitude of a displacement of the normal axis of rotationof a rotationally vibrated rotary body, the displacement correspondingto the position and amount of the unbalance of said rotary body. Therotary body is rotationally vibrated within a minute angular rangearound its own normal axis of rotation and is resiliently supported totwo spaced points along said axis so that it may be displaced in a planeperpendicular to said axis. As a result, said normal axis of rotation ofthe rotary body is displaced as a function of the angular position andamount of the unbalance of the rotary body. The magnitude and directionof the displacement are detected, whereby the angular position andamount of the dynamic unbalance of the rotary body are de tected.

8 Claims, 5 Drawing Figures PATENTEUmm 31315 7 0,11 0

' SHEET 10F 2 sum 20F 2 METHOD AND APPARATUS FOR DETECTING ANGULARPOSITION AND AMOUNT OF DYNAMIC UNBALANCE OF ROTATING BODY BACKGROUND OFTHE INVENTION The present invention relates to apparatus for detectingthe angular position and amount of a dynamic unbalance of a rotary body.

If a rotary body such as, for example, a rotor in an electric motor hasa dynamic unbalance, the centrifugal whirling of the shaft is caused atrotation, so that no stable rotation is attainable. In addition,breakage of the rotary shaft and/or damage of the rotor itself or theperipheral equipments are also brought about. Therefore, the rotary bodyisrequired to have no dynamic unbalance. However, it is extremelydifficult to manufacture the rotary body including no unbalance at themanufacturing step thereof. Accordingly, the general procedure has beenthat, after the manufacture of the rotary body, a part of the rotarybody is cut off or a small piece is fixed to a part of the rotary bodyin dependence upon the position and amount of the dynamic unbalance ofthe rotary body, thereby removing the dynamic unbalance. In this case,it is very important to know the amount and position of the dynamicunbalance of the manufactured rotary body.

A known apparatus for detecting the position and amount of an unbalanceof a rotary body as are required for the correction of the dynamicbalance of the rotary body, comprises means to rotate the rotary body ata constant speed, means to detect in the form of signals the centrifugalwhirling of the shaft due to a dynamic unbalance of the rotary bodyduring its rotation, reference-signal generator means, and means to makecomparisons between the detected signal and the reference signal,whereby the angular position and amount of the dynamic unbalance of therotary body are detected.

To know the position and amount of the dynamic unbalance of the rotarybody by rotating it at a constant speed, is a preferable detectingmethod since the dynamic unbalance can be detected in the same conditionas the actual state of use of the rotary body. Since, however, thedetector means for the centrifugal whirling of the shaft, thereference-signal generator means, and the means for carrying out thecomparison between the detected signal and the reference signal arerequired, the apparatus is complicated. Moreover, as regards a rotarybody susceptible to air resistance during rotation, such as a fan, onelarge in miscellaneous vibrations, such as an anti-friction bearing, orone large in deflection, the detection of the dynamic unbalance has beendifficult with the detecting method as described above.

OBJECTS OF THE INVENTION:

It is accordingly an object of the present invention to provide a methodand/or apparatus for detecting the angular position and amount of adynamic unbalance without rotating a rotary body at a constant speed.

A further object of the present invention is to provide a method andapparatus according to which the angular position and amount ofa dynamicunbalance of a rotary body may be detected without any step ofprocessing electrical signals.

0 amount of a dynamic unbalance of a rotary body,

which apparatus is simple in construction and in handling and is low incost.

SUMMARY OF THE INVENTION According to the present invention, the angularposition and amount ofa dynamic unbalance of rotary body is detected byresiliently supporting the rotary body at at least two spaced pointsalong the normal axis of rotation of the rotary body so that the supportpoints are resiliently restrained for motion in any direction in planestransverse to the axis. Rotational vibration within a minute angularrange about the normal axis of rotation of the rotary body is impartedto the rotary body so that the rotary body causes the generation of thedisplacement of the normal axis of displacement thereof due to thedynamic unbalance thereof. The displacement of the normal axis ofrotation of the rotary body is detected to thereby detect the amount andangular position of the dynamic unbalance of the rotary body which is afunction of the displacement. The present invention contemplates boththe method and apparatus for carrying out the above inventive concept.

Herein, the expression the rotational vibration within a predeterminedminute angular range means a repetitive motion consisting of forwardrotations of the chucks over a minute angle and backward rotations bythe same angle as in the forward rotations.

Further features, objects and concrete constructions for realizing them,of the present invention will be readily understood from the followingdescription of the preferred embodiments of the invention taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. l is a schematic view forexplaining the operating principle of apparatus of the presentinvention;

FIG. 2 is a front view of an apparatus embodying the present invention;

FIG. 3 is a sectional view showing the construction of bearing andvibrating means which is used in the apparatus shown in FIG. 2;

FIG. 4 is a combined sectional view taken along line a a and line b b inFIG. 3, the left half of the figure showing a sectional view along theline a a and the right halfa sectional view along the line b b; and

FIG. 5 is a view showing a reference mark and a scale plate.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring to FIG. I, adisc I has an unbalance of a mass m (which is termed the staticunbalance) in a place at a distance r from the center thereof. Then, themean value F of a centrifugal force exerted upon the unbalanced portionof the disc I" when the disc is rotationally vibrated within a minuteangular range about its axis, is calculated as follows:

An angular amplitude at the unbalanced part is 0 6,, sin wt (1) where0,, represents the maximum angular amplitude, while w the angularvelocity which is expressed by 21rf if the frequency is assumed to bef.t indicates the time.

On the other hand, sice the unbalanced part is conducting a circularmotion at an optional time during the vibration, a centrifugal force Facting upon the unbalanced portion may be expressed, on the assumptionthat the angular velocity at that time be a, as below.

F m r a (2) Since, on the other hand, the angular velocity a is given asthe differentiated value of 0 in expression (1) with respect to timeexpression (2) is transformed as follows:

F m r (dB/d!) m r [(d/dt) (O Sin Wt)] m r (6,,W cos Wt) =m r 0 W cos wtAccordingly, the average value F of this centrifugal force F= JL Fdt= JLmrfl W cos wtdt Herein, T is the period and T l/f= 21r/w, so that F=(mr6 w I)'% T A m r w 0,, 3 If the maximum amplitude 0 is made small, thedirection in which the average centrifugal force F acts is fixed, ineffect, to the direction which connects the center of the vibration ofthe unbalanced part and the rotational center or the center 0, of thedisc P,.

Accordingly, if the rotational vibration as described above is impartedto the disc P,, the center axis of the disc P, is displaced in thedirection in which there is the unbalance and by an amount whichcorresponds to the magnitude of the unbalance. Therefore, the angularposition and the amount of the unbalance may be detected by observingthe displacement of the center axis.

In order to allow the displacement of the center axis, the rotary shaftof the disc P, may be resiliently supported by springs or the like, inFIG. I, the supporting means are illustrated as being springs S to S andS to S Also shown in FIG. 1 is a disc P as to which the position andamount of an unbalance thereof may be detected from the direction andmagnitude of the displacement of the dislocation or deviation of theaxis as in the above-mentioned disc P,.

An actual rotary body has a thickness in its axial direction, and may beconsidered as one in which innumerable discs each having a staticunbalance are closely arranged. It is accordingly deemed that, in anactual rotary body, the dislocations of axes due to the innumerableunbalances arranged in the axial direction are composed when it isrotated, thus equivalently produc ing the static unbalance at the rightand left side parts of the rotary body. The correction of the dynamicbalance may be done by eliminating the equivalent static unbalance.

F= m r w (4) The number of revolutions N per minute in this case is asbelow, the angular velocity and the period of rotation beingrespectively indicated by w andf.

N=60f= 60 (w/21r) From expressions (4) and (5 N=60l21r= V F mr (6) Ingeneral, if the centrifugal force F when a mass m at a point distant byr from the center is rotated about the center is known, its number ofrevolutions per minute may be calculated from expression (6).

Accordingly, by substituting F in the previous expression (3) into F inexpression (6), the equivalent number of revolutions N when the disc isrotationally vibrated at the frequency off and the angular amplitude of0 may be evaluated. That is,

60 1 =5 5 mrw 0 Assuming now that f= 300Hz and 0,, 0.1rad, the

mean centrifugal force Fand the number of revolutions N are calculatedfrom expressions (3) and (7):

Now, if the disc P, is'rotated so as to produce the same centrifugalforce as in expression (8), its number of revolutions N and itsrotational frequencyfare given from expression (4) and from expression(5) This value of N is equal to the equivalent number of revolutions inthe case where the disc is rotationally vibrated.

Accordingly, it will be understood from this fact that the rotationalvibration of the disc within a minute angular range around the axis andthe rotation of the disc are regarded as being equivalent in thedetection of the unbalance.

On the other hand, in case of the rotational vibration, the direction ofthe centrifugal force due to the unbalance, accordingly the direction ofthe displacement of the axis varies only within the minute angularrange, so that the direction may be substantially regarded as beingstationary.

It will accordingly be understood that a dynamic unbalance of a rotarybody is statically measured as the displacement of its axis byrotationally vibrating the rotary body within a minute angular rangeabout said axis.

Referring to FIG. 2, which shows a front view of an embodiment in whichthe present invention is applied to the case of detecting the positionand amount of a dynamic unbalance of a relatively small-sized rotor, theapparatus comprises a base 1, frames 2 and 2' vertically raised from theright and left ends (as seen in the figure) of the base 1 and fixed tothe base 1, two driving devices 3 and 3 mounted on the base 1 and therespective frames 2 and 2', chucks 4 and 4 driven by the respectivedriving devices and for driving a rotary body R, and unbalance detectingdevices and 5'.

The chucks 4 and 4' oppose to each other, and are arranged at suchpositions that their center axes are located on an identical axis, thatthe rotary body R may be inserted therebetween, and that the rotary bodyR may be supported shafts R, and R thereof. The chucks 4 and 4' arefixed to driving shafts (34 in FIGS. 3 and 4) of the driving devices 3and 3', respectively, while the driving devices 3 and 3' are elasticallysupported in such manner that the driving shafts are normally located onan identical axis.

Accordingly, when the driving devices 3 and 3' are driven torotationally vibrate the rotary body R within a minute angular rangeabout its rotary axis, the shafts R, and R conduct the displacement, onthe basis of the foregoing principle, due to the dynamic unbalance ofthe rotary body and in conformity with the magnitude and the directionof the angular position of the unbalance. Since the shafts R, and R arecoupled to the driving shafts by means of the chucks 4 and 4' and thedriving devices having the driving shafts are resiliently supported, aforce to displace the shafts R, and R becomes a force to displace thedriving devices 3 and 3', and the driving devices 3 and 3' are displaceduntil this force and the resilient supporting force are balanced witheach other. Accordingly, the position and amount of the dynamicunbalance of the rotary body R may be determined by detecting thedisplacement by means of the detecting devices 5 and 5'.

It will be easily understood from the foregoing explanation that, as thedetecting devices 5 and 5', there may be used various detecting devicesfor a static force or for a displacement, such as a spring balancer, a

parallel spring mechanism, and a combination of a strain gauge and anRC, and a oscillator the strain gauge serving as the resistive (R)element of the oscillator to vary the frequency of the oscillator as afunction of the strain applied to the strain gauge.

Referring now to FIGS. 3 and 4, description will be made of theconstruction of an embodiment of the driving device. Diagrammaticallyshown in the figures are the construction for the resilient support, thedetecting device, and the construction of the chuck.

The driving device comprises a cylindrical housing 31 closed at one endand open at the other end, an electromagnet in which a horse-shoe-shapedyoke 32 made of a magnetic substance has a winding 33 wound thereon, adriving shaft 34 which extends along the center axis between two legs ofthe yoke 32, four leaf springs 35 for holding the driving shaft 34 atits position, to spring bearings 36 for supporting two of the leafsprings, and a magnetic-substance piece 37 made of such material as ironwhich is fixed to the driving shaft 34 and which has a lengthsubstantially equal to the length across the legs of the yoke 32.

Each of the flat springs 35 has one end fixed to the driving shaft 34and has the other end fixed to the leg of the yoke 32 or the springbearing 36, with the result that the driving shaft extends at suchposition. The iron piece 37 fixed to the driving shaft 34 is secured ina manner to be dislocated by a minute angle 0 around the center axis ofthe driving shaft 34 with respect to a line connecting the leg portionsof the yoke 32.

When a current flows through the winding 33, a magnetic flux isgenerated through the yoke 32. As a result, the iron piece 37 is turnedin the direction of closing a magnetic circuit through which themagnetic flux passes, until both ends of the iron piece oppose the endfaces of the legs of the yoke 32. Herein, if the current to the winding33 is interrupted, the iron piece returns to the original position bythe elasticity of the leaf springs 35. Therefore, when an alternatingcurrent is applied to the coil 33, the iron piece 37 and accordingly thedriving shaft 34 are rotationally vibrated within the range of theminute angle 0 about its center axis.

To the driving shaft 34, the chuck 4 is fixed. The chuck 4 is secured tothe driving shaft 34 such that its center is located on the center axisof the driving shaft 34, while it grips the shaft R, of the rotary bodyand thus supports the rotary body so as to make the center axis of therotary body and that of the driving shaft 34 coincide with each other.

Since the chuck 4 has hitherto been widely known in a variety ofmechanical devices, the detailed explanation is omitted herein. Amongchucks having heretofore been used, there are included various ones suchas the independent chuck, the universal chuck, and the like. Any chuckmay be employed.

The housing 31 is resiliently supported by, e. g., four springs 6arranged on an identical plane, with the result that it may be displacedin optional directions within the plane (within a plane parallel to thedrawing as viewed in FIG. 4).

On the other hand, a reference mark 7 is provided at a part of theclosed end face of the housing 31, while a flat transparent orsemi-transparent scale plate 8 is disposed in a manner to oppose to saidend face. As shown in FIG. 5, the scale plate 8 has a plurality ofconcentric circles 82 with respect to a central point 81 and straightlines 83 radially extending from the central point 81. It is graduatedsuch that the concentric circles indicate a displacement or force, orthe magnitude of a dynamic unbalance, while the straight lines 83represent the angular scale, and thus, it is previously adjusted so asto cause the central point 81 and the reference mark 7 to coincide.Herein, when the rotary body R is rotationally vibrated by driving thedriving device 3, it is intended to conduct a displacement in dependenceon its dynamic unbalance as has been stated above. As a result, thedriving device 3 is displaced against the elasticity of the springs 6.Accordingly, the position and amount of the dynamic unbalance may bedetermined by reading a scale on the scale plate 8 at the position towhich the reference mark 7 has moved, i.e., a graduation indicated bythe concentric circles 82 and a graduation indicated by the straightlines 83.

it will be readily understood that, while the detecting device 5 hasbeen composed of the reference mark 7 and the scale plate 8, it mayotherwise be constructed such that e. g. the extension and contractionof the respective springs 6 are detected by means of, for example, astrain gauge to evaluate the position and magnitude of the dynamicunbalance from the detected values.

While, in the foregoing, the present invention has been described of thespecific embodiments thereof, it will be easily understood that theinvention is not restricted to the described embodiments, but that avariety of designs and modifications may be made within the scope of thepresent invention.

What is claimed: 1. A method for detecting the angular position andamount of a dynamic unblance of a rotary body, com prising the steps of:

resiliently supporting the rotary body at at least two spaced pointsalong the normal axis of rotation of the rotary body so that the atleast two support points are resiliently restrained for motion in anydirection in planes transverse to said axis;

imparting to said rotary body a rotational vibration within a minuteangular range about the normal axis of rotation thereof, so that saidrotary body is caused to generate a displacement of said normal axis ofrotation due to the dynamic unbalance thereof; and detecting themagnitude and direction of said displacement of said normal axis ofrotation of said rotary body to thereby detect the amount and angularposition of the dynamic unbalance of said rotary body. 2. Apparatus fordetecting the angular position and amount of a dynamic unbalance of arotary body comprising:

means for resiliently supporting said rotary body at at least two spacedpoints along the normal axis of rotation of said rotary body so that theat least two support points are resiliently restrained for motion in anydirection in planes transverse to said axis;

means to impart to said rotary body a rotational vibration within aminute angular range about the normal axis of rotation thereof, so thatsaid rotary body is caused to generate a displacement of said normalaxis of rotation due to the dynamic unbalance thereof; and

means to detect the magnitude and direction of said displacement of saidnormal axis of rotation of said rotary body to thereby detect the amountand angular position of the dynamic unbalance of said rotary body whichis a function of said displacement.

3. Apparatus for detecting the angular position and amount of a dynamicunbalance of a rotary body comprising:

a base;

two driving devices mounted on said base in opposing relation to eachother, said driving devices carrying said rotary body at two spacedpoints along the normal axis of rotation of said rotary body andimparting rotational vibration to said rotary body;

resilient support means for resiliently mounting each of said twodriving devices to said base so that said driving devices and saidrotary body are resiliently restrained for motion in any direction inplanes transverse to said axis; and

means to detect displacement of the resiliently supported drivingdevices caused by the rotational vibration of said rotary body tothereby detect the angular position of said rotary body which is afunction of said displacements.

4. Apparatus according to claim 3 comprising chuck means on said drivingdevices for gripping said rotary body and mounting said rotary body tosaid driving means.

5. Apparatus according to claim 4 wherein each of said two drivingdevices comprises:

a cylindrical housing closed at one end and opened at the other end;

an electromagnet including a horse-shoe-shaped yoke received in saidhousing and a coil wound thereon, a separated part of said yoke beinglocated at the open end of said housing;

a driving shaft extending on the center axis of said housing andexternally protruding out of said open end of said housing;

resilient support means for fixing said driving shaft in position andfor supporting said driving shaft such that said driving shaft iscapable of being driven about its own axis by an external force and isreturnable to its original position when said external force is removed,said resilient support means comprising a plurality of spring means,each spring means having one end fixed outside said housing of said eachdriving devices, each spring means extending substantially in the radialdirection of said housing, and each spring means having the other endfixed to said base; and

a magnetic-substance piece fixed to the protruding part of said drivingshaft and extending in the radial direction of said housing, saidmagnetic-substance piece being deviated from said yoke by a minute anglearound said center axis of said housing while said electromagnet is notenergized, in order to prevent said separated part of said yoke frombeing short circuited;

each of said chucks being fixed to the protruding end of said drivingshaft of the corresponding one of said two driving devices; and

said displacement detecting means comprising means for detecting theamount of change of each of said plurality of spring means and foroperating on said amounts thereby detecting to determine lines forindicating the angle of the displacement, the deviation of saidreference mark from the central position of said scale plate indicatingthe angular position and amount of the dynamic unbalance.

7. Apparatus as claimed in claim 6 wherein said scale plate istransparent.

8. Apparatus as claimed in claim 6 wherein said scale plate issemi-transparent.

1. A method for detecting the angular position and amount of a dynamicunblance of a rotary body, comprising the steps of: resilientlysupporting the rotary body at at least two spaced points along thenormal axis of rotation of the rotary body so that the at least twosupport points are resiliently restrained for motion in any direction inplanes transverse to said axis; imparting to said rotary body arotational vibration within a minute angular range about the normal axisof rotation thereof, so that said rotary body is caused to generate adisplacement of said normal axis of rotation due to the dynamicunbalance thereof; and detecting the magnitude and direction of saiddisplacement of said normal axis of rotation of said rotary body tothereby detect the amount and angular position of the dynamic unbalanceof said rotary body.
 1. A method for detecting the angular position andamount of a dynamic unblance of a rotary body, comprising the steps of:resiliently supporting the rotary body at at least two spaced pointsalong the normal axis of rotation of the rotary body so that the atleast two support points are resiliently restrained for motion in anydirection in planes transverse to said axis; imparting to said rotarybody a rotational vibration within a minute angular range about thenormal axis of rotation thereof, so that said rotary body is caused togenerate a displacement of said normal axis of rotation due to thedynamic unbalance thereof; and detecting the magnitude and direction ofsaid displacement of said normal axis of rotation of said rotary body tothereby detect the amount and angular position of the dynamic unbalanceof said rotary body.
 2. Apparatus for detecting the angular position andamount of a dynamic unbalance of a rotary body comprising: means forresiliently supporting said rotary body at at least two spaced pointsalong the normal axis of rotation of said rotary body so that the atleast two support points are resiliently restrained for motion in anydirection in planes transverse to said axis; means to impart to saidrotary body a rotational vibration within a minute angular range aboutthe normal axis of rotation thereof, so that said rotary body is causedto generate a displacement of said normal axis of rotation due to thedynamic unbalance thereof; and means to detect the magnitude anddirection of said displacement of said normal axis of rotation of saidrotary body to thereby detect the amount and angular position of thedynamic unbalance of said rotary body which is a function of saiddisplacement.
 3. Apparatus for detecting the angular position and amountof a dynamic unbalance of a rotary body comprising: a base; two drivingdevices mounted on said base in opposing relation to each other, saiddriving devices carrying said rotary body at two spaced points along thenormal axis of rotation of said rotary body and imparting rotationalvibration to said rotary body; resilient support means for resilientlymounting each of said two driving devices to said base so that saiddriving devices and said rotary body are resiliently restrained formotion in any direction in planes transverse to said axis; and means todetect displacement of the resiliently supported driving devices causedby the rotational vibration of saiD rotary body to thereby detect theangular position of said rotary body which is a function of saiddisplacements.
 4. Apparatus according to claim 3 comprising chuck meanson said driving devices for gripping said rotary body and mounting saidrotary body to said driving means.
 5. Apparatus according to claim 4wherein each of said two driving devices comprises: a cylindricalhousing closed at one end and opened at the other end; an electromagnetincluding a horse-shoe-shaped yoke received in said housing and a coilwound thereon, a separated part of said yoke being located at the openend of said housing; a driving shaft extending on the center axis ofsaid housing and externally protruding out of said open end of saidhousing; resilient support means for fixing said driving shaft inposition and for supporting said driving shaft such that said drivingshaft is capable of being driven about its own axis by an external forceand is returnable to its original position when said external force isremoved, said resilient support means comprising a plurality of springmeans, each spring means having one end fixed outside said housing ofsaid each driving devices, each spring means extending substantially inthe radial direction of said housing, and each spring means having theother end fixed to said base; and a magnetic-substance piece fixed tothe protruding part of said driving shaft and extending in the radialdirection of said housing, said magnetic-substance piece being deviatedfrom said yoke by a minute angle around said center axis of said housingwhile said electromagnet is not energized, in order to prevent saidseparated part of said yoke from being short circuited; each of saidchucks being fixed to the protruding end of said driving shaft of thecorresponding one of said two driving devices; and said displacementdetecting means comprising means for detecting the amount of change ofeach of said plurality of spring means and for operating on said amountsthereby detecting to determine the direction and magnitude of thedisplacement of said driving devices.
 6. Apparatus as claimed in claim 5wherein said displacement detecting means comprises a reference markprovided on the outer surface of said closed end of said housing of eachof said driving devices and a scale plate fixed to said base opposed tosaid closed end, said scale plate comprising a plurality ofconcentrically circular scale lines for indicating the amount ofdisplacement and radially directed straight linear scale lines forindicating the angle of the displacement, the deviation of saidreference mark from the central position of said scale plate indicatingthe angular position and amount of the dynamic unbalance.
 7. Apparatusas claimed in claim 6 wherein said scale plate is transparent.